Does there exist a sequence of complex numbers $s = \left( s_1, s_2, \cdots \right)$ which is exponentially bounded, i.e. there are $C, \tau > 0$ such that $|s_n| \leq C e^{\tau n}$ for all $n \in \{1,2,\cdots\}$, but for which no function $f: \{ \text{Re}(z) \geq 0 \} \rightarrow \mathbb{C}$ exists which satisfies the following properties:
$f$ takes on the values in $s$ on the positive integers, i.e. $f(n) = s_n$ for all $n \in \{1,2,\cdots\}$,
$f$ is analytic in the right half-plane $\text{Re}(z) > 0$ and continuous onto the imaginary axis,
$f$ is exponentially bounded in the right half-plane, i.e. there are $C', \tau' > 0$ such that $|f(z)| \leq C' e^{\tau' |z|}$ for all $z$ with $\text{Re}(z) > 0$,
there are $C'' > 0$ and $\tau'' < \pi$ such that on the imaginary axis $f$ is bounded as $|f(iy)| \leq C'' e^{\tau''|y|}$?
This question is inspired by Carlson's theorem which states that if there is a function $f$ that satisfies the above properties with $s = (0,0,\cdots)$ the zero-sequence, then $f$ is identically zero in the right half-plane plus imaginary axis. I am wondering whether there are cases where all the analytic continuations of a function defined on the positive integers via $f(n) = s_n$ to the right half-plane necessarily violate the conditions that appear in Carlson's theorem.
A bonus question: are there also such sequences $s$ containing only positive real numbers?